DSA Notes
HubAlgorithm vs Data Structure — Key Differences, Definitions, and How They Work TogetherApplications of DSA — How Data Structures and Algorithms Power Real-World SoftwareTime and Space Complexity — A Complete BeginnerWhat is DSA? — A Complete BeginnerWhy Learn DSA? — Career Benefits, Real-World Impact, and Long-Term ValueAsymptotic Analysis — Complete Framework with Recurrence Relations and Master TheoremBest, Average, and Worst Case Analysis — A Practical Guide for Every AlgorithmBig O Notation — Complete Guide with Formal Definition, Rules, and ExamplesBig Omega Notation — Lower Bounds, Best Case, and Formal DefinitionBig Theta (Θ) Notation — Tight Bounds and Exact Growth RatesSpace Complexity — Memory Analysis, Auxiliary Space, and Time-Space TradeoffsTime Complexity Examples — 25 Worked Problems from Basic to AdvancedArray Interview Problems — 15 Classic Problems with Complete SolutionsArray Operations — Every Operation Explained with Code, Complexity, and Edge CasesArray Rotation — Algorithms, Techniques, and Interview ProblemsArray Searching — Linear Search, Binary Search, and Advanced TechniquesArrays — The Complete Introduction to the Most Fundamental Data StructureDynamic Arrays — How Python Lists and ArrayLists Work InternallyMultidimensional Arrays — 2D Arrays, Matrices, and Memory LayoutSliding Window Technique — Complete Guide with 10 Classic ProblemsAnagram Problems — Detection, Grouping, and Sliding Window VariationsPalindrome Problems — Check, Longest, Count, and Expand Around CenterPattern Searching — KMP, Rabin-Karp, and Boyer-Moore AlgorithmsString Interview Problems — 10 Classic Problems with Complete SolutionsString Matching — Longest Common Subsequence, Edit Distance, and Z-AlgorithmString Operations — Complete Reference with Complexity and ImplementationStrings — Complete Introduction to String Data Structure and OperationsCircular Linked List — Structure, Operations, and ApplicationsDoubly Linked List — Complete Implementation with Bidirectional OperationsLinked List Interview Problems — 12 Classic Problems with Complete SolutionsLinked Lists — A Complete Introduction with Memory Model and TypesLinked List Operations — Every Technique with Code and ComplexityReverse Linked List — All Approaches from Basic to AdvancedSingly Linked List — Complete Implementation with All OperationsBalanced Parentheses — Stack-Based Validation with All VariantsInfix, Prefix, and Postfix Expressions — Stack-Based Conversion and EvaluationStack Implementation — Three Complete Implementations with Edge CasesStack Interview Problems — 10 Classic Problems with Complete SolutionsStack — Complete Introduction to LIFO Data StructureStack Using Arrays — Memory Layout, Capacity Management, and PerformanceStack Using Linked List — Implementation, Analysis, and When to Use ItCircular Queue — The Ring Buffer Explained with ImplementationDeque — Double-Ended Queue with Complete Implementation and ApplicationsPriority Queue — Complete Guide from Concept to ImplementationQueue Applications — BFS, Level-Order Traversal, 01-BFS, and MoreQueue Implementation — Array, Linked List, and Two-Stack ApproachesQueue Interview Problems — 10 Classic Problems with Complete SolutionsQueue — Complete Introduction to the FIFO Data StructureAVL Tree - Self-Balancing Binary Search TreeBinary Search Tree (BST)Binary Tree - Structure and OperationsFenwick Tree (Binary Indexed Tree)Red-Black Tree - Another Balanced BSTSegment Tree - Range Query Data StructureTree Interview Problems CollectionTree Data Structure FundamentalsTree Traversals - Inorder, Preorder, Postorder, Level OrderHeap — Complete Introduction to the Priority-Based Tree Data StructureHeap Operations — Complete Reference with Complexity AnalysisHeap Problems — 8 Classic Interview Problems with Complete SolutionsMax Heap — Complete Implementation and ApplicationsMin Heap — Complete Implementation with All OperationsPriority Queue Using Heap — Implementation, Applications, and PatternsBellman-Ford Algorithm — Shortest Path with Negative Weights and Cycle DetectionBFS Algorithm — Breadth-First Search with All Applications and VariantsDFS Algorithm — Depth-First Search with Cycle Detection and All ApplicationsDijkstraFloyd-Warshall Algorithm — All-Pairs Shortest Paths with Complete AnalysisGraph Interview Problems — 12 Classic Problems with Complete SolutionsGraph — Complete Introduction to the Most Powerful Data StructureGraph Representation — Adjacency List, Matrix, Edge List with Complete AnalysisKruskalMinimum Spanning Tree — Definition, Properties, and ApplicationsPrimShortest Path Algorithms — Complete Comparison and Selection GuideTopological Sorting — DFS and BFS (KahnUnion-Find (DSU) in Graphs — Connected Components and Dynamic ConnectivitySeparate Chaining — Complete Implementation and AnalysisCollision Handling — Chaining vs Open Addressing Complete ComparisonHash Functions — Design, Properties, and Common ImplementationsHashing — Complete Introduction to Hash Tables and Hash FunctionsHashing Problems — 10 Classic Interview Problems with Complete SolutionsOpen Addressing — Linear Probing, Quadratic Probing, and Double HashingBubble Sort — Complete Analysis with Optimizations and When to Use ItCounting Sort — O(n+k) Sorting That Breaks the Comparison Lower BoundHeap Sort — Guaranteed O(n log n) with O(1) SpaceInsertion Sort — The O(n) Best Case Sort That Powers PythonMerge Sort — Complete Guide to the Guaranteed O(n log n) Stable SortQuick Sort — The Fastest Practical Sorting Algorithm with Complete AnalysisRadix Sort — O(nk) Digit-by-Digit Sorting for Large Integer RangesSelection Sort — Complete Analysis with Swaps Minimization PropertySorting Algorithm Comparison — The Complete Decision GuideSorting — Complete Introduction to Sorting AlgorithmsBinary Search — Complete Guide with All Variants and Proof of CorrectnessInterpolation Search — O(log log n) Search for Uniformly Distributed DataLinear Search — Complete Guide with All Variants and ApplicationsSearch Algorithm Comparison — Choosing the Right Search for Every SituationSearching Interview Problems — 10 Classic Problems with Complete SolutionsTernary Search — Finding Maximum/Minimum of Unimodal FunctionsBacktracking Introduction — The Recursive Exploration with UndoRecursion Basics — The Complete BeginnerRecursion Problems — 12 Classic Problems from Basic to AdvancedRecursion Tree — Visualizing Recursion for Complexity AnalysisRecursion vs Iteration — When to Use Each and How to ConvertTail Recursion — Definition, Optimization, and Converting to IterationDP Interview Problems — 12 Classic Problems with Pattern Recognition GuideDynamic Programming — Complete Introduction with Properties and Problem IdentificationFibonacci DP — From Exponential to Linear with Memoization and Matrix ExponentiationKnapsack Problem — 0/1, Unbounded, and Fractional with Complete AnalysisLongest Common Subsequence — Complete Guide with DP Table and String VariantsLongest Increasing Subsequence — O(n log n) with Patience SortingMatrix Chain Multiplication — Interval DP with Complete AnalysisMemoization — Top-Down Dynamic Programming with CachingTabulation — Bottom-Up Dynamic Programming with Complete Table BuildingActivity Selection Problem — Greedy Scheduling with Proof of CorrectnessFractional Knapsack — Greedy by Value Density with ProofGreedy Interview Problems — 10 Classic Problems with Complete SolutionsGreedy Algorithms — Complete Introduction with Proof TechniquesHuffman Coding — Optimal Prefix-Free Data CompressionJob Sequencing Problem — Greedy Scheduling with DeadlinesBacktracking — The Complete Guide to Recursive Exploration with UndoBacktracking Problems — 12 Classic Interview Problems with Complete SolutionsN-Queens Problem — Complete Solution with Backtracking and OptimizationRat in a Maze — Backtracking Path Finding with All SolutionsSubset Generation — All Subsets, Combinations, and Partitions with BacktrackingSudoku Solver — Backtracking with Constraint PropagationCodeforces Guide — Complete Strategy to Reach 1800+ RatingCommon CP Techniques — Essential Algorithms Every Competitive Programmer Must KnowContest Strategies — How to Perform Your Best in Competitive Programming ContestsCompetitive Programming — Complete Introduction for BeginnersFast Input/Output — Why I/O Speed Matters and Complete Templates for C++ and PythonLeetCode Guide — Complete Strategy to Crack FAANG InterviewsBit Manipulation — Complete Guide with All Tricks and Interview ProblemsBloom Filter — Space-Efficient Probabilistic Membership TestingComputational Geometry — Points, Lines, Convex Hull, and Geometric AlgorithmsDisjoint Set Union (DSU) — Union-Find with Path Compression and Union by RankRandomized Algorithms — Las Vegas, Monte Carlo, and Probabilistic TechniquesSkip List — The Probabilistic Data Structure that Competes with Balanced BSTsSuffix Array — Construction, LCP Array, and String Search ApplicationsTrie Data Structure — Complete Guide with Implementation and ApplicationsAutocomplete System Using TrieLibrary Management System Using DSARoute Finder Using GraphsMini Search Engine ProjectSocial Network Analysis Using GraphsEssential Coding Patterns for InterviewsDSA Cheatsheet: Quick ReferenceEssential DSA Interview QuestionsFAANG DSA Sheet: Company-Specific Patterns and ProblemsProblem Solving Strategies for Coding InterviewsComplexity CheatsheetImportant Algorithms SummaryBest Practice Platforms for DSARecommended Books for DSA
Master the balanced parentheses problem — validating matching brackets using a stack, all variants (multiple bracket types, minimum additions to balance, longest valid substring), why stacks are the perfect data structure for bracket matching, and interview patterns.
Problem Definition
A string of brackets is balanced if:
- Every opening bracket has a matching closing bracket
- Brackets close in the correct order (LIFO — last opened, first closed)
Example
Valid: () []{} {[()]} ((()))
Invalid: (] ([)] (() {[}
Implementation — Validate Brackets
Example
def is_valid(s):
"""
Check if string of brackets is valid.
Time: O(n), Space: O(n)
"""
stack = []
matching = {')': '(', ']': '[', '}': '{'}
for char in s:
if char in '([{':
stack.append(char) # Push opening brackets
elif char in ')]}':
if not stack or stack[-1] != matching[char]:
return False # No match
stack.pop() # Matching bracket found
return len(stack) == 0 # Valid only if all openers were matched
print(is_valid("()[]{}")) # True
print(is_valid("([{}])")) # True
print(is_valid("(]")) # False
print(is_valid("([)]")) # False — wrong order
print(is_valid("")) # True — empty is valid
Variant 1 — Minimum Additions to Make Valid
python example
def min_add_to_make_valid(s):
"""
Return minimum number of bracket additions to make s valid.
Count unmatched opens and closes.
"""
open_count = 0 # Unmatched '('
close_count = 0 # Unmatched ')'
for char in s:
if char == '(':
open_count += 1
else: # char == ')'
if open_count > 0:
open_count -= 1 # Matched with an open
else:
close_count += 1 # Unmatched close
return open_count + close_count # Need one addition per unmatched
print(min_add_to_make_valid("())")) # 1
print(min_add_to_make_valid("(((")) # 3
print(min_add_to_make_valid("()")) # 0Variant 2 — Longest Valid Parentheses Substring
Example
def longest_valid_parentheses(s):
"""
Find length of longest valid parentheses substring.
Stack-based: track indices.
Time: O(n), Space: O(n)
"""
stack = [-1] # Sentinel: index before start of current valid sequence
max_len = 0
for i, char in enumerate(s):
if char == '(':
stack.append(i) # Push opening bracket index
else: # ')'
stack.pop() # Pop matching opener (or sentinel)
if not stack:
stack.append(i) # New base (unmatched ')')
else:
max_len = max(max_len, i - stack[-1])
return max_len
print(longest_valid_parentheses(")()())")) # 4 — "()()"
print(longest_valid_parentheses("(()")) # 2 — "()"
print(longest_valid_parentheses("")) # 0
Variant 3 — Score of Parentheses
Example
def score_of_parentheses(s):
"""
Calculate score: '()' = 1, outer brackets double inner score.
Stack tracks running totals at each nesting level.
"""
stack = [0] # Start with 0 score at level 0
for char in s:
if char == '(':
stack.append(0) # New level, start at 0
else:
v = stack.pop()
stack[-1] += max(2 * v, 1) # '()' = 1, outer doubles inner
return stack[0]
print(score_of_parentheses("()")) # 1
print(score_of_parentheses("(())")) # 2
print(score_of_parentheses("()()")) # 2
print(score_of_parentheses("(()(()))")) # 6
Variant 4 — Generate All Valid Combinations
A related problem: generate all valid bracket combinations of length 2n.
Example
def generate_parentheses(n):
"""
Generate all valid parentheses combinations of n pairs.
Backtracking: add '(' if open count < n, add ')' if close < open.
Time: O(4^n / sqrt(n)) — Catalan number, Space: O(n) recursion depth
"""
result = []
def backtrack(current, open_count, close_count):
if len(current) == 2 * n:
result.append(current)
return
if open_count < n:
backtrack(current + '(', open_count + 1, close_count)
if close_count < open_count:
backtrack(current + ')', open_count, close_count + 1)
backtrack('', 0, 0)
return result
print(generate_parentheses(3))
# ['((()))', '(()())', '(())()', '()(())', '()()()']
The number of valid combinations for n pairs is the n-th Catalan number: C(n) = (2n choose n) / (n+1). For n=3, C(3) = 5, which matches the output above.
Variant 5 — Remove Invalid Parentheses
Given a string with letters and brackets, remove the minimum number of brackets to make it valid. Return all unique results.
Example
def remove_invalid_parentheses(s):
"""
BFS approach: try removing each bracket at each level.
Stop at first level where valid strings are found.
Time: O(2^n), Space: O(2^n) in worst case
"""
from collections import deque
def is_valid(t):
count = 0
for c in t:
if c == '(': count += 1
elif c == ')':
count -= 1
if count < 0: return False
return count == 0
visited = {s}
queue = deque([s])
result = []
found = False
while queue:
level_size = len(queue)
for _ in range(level_size):
cur = queue.popleft()
if is_valid(cur):
result.append(cur)
found = True
if not found:
for i in range(len(cur)):
if cur[i] in '()':
nxt = cur[:i] + cur[i+1:]
if nxt not in visited:
visited.add(nxt)
queue.append(nxt)
if found:
break
return result if result else ['']
print(remove_invalid_parentheses("()())()")) # ['()()()', '(())()']
Interview Patterns and Tips
Pattern 1: Stack with Extra Data
Sometimes you need to store more than the bracket itself. Store the index for problems asking about the position of invalid brackets:
Example
def find_invalid_positions(s):
"""Return positions of invalid brackets."""
stack = [] # Store indices of unmatched '('
invalid = set()
for i, char in enumerate(s):
if char == '(':
stack.append(i)
elif char == ')':
if stack:
stack.pop() # Matched
else:
invalid.add(i) # Unmatched ')'
# Remaining in stack are unmatched '('
invalid.update(stack)
return sorted(invalid)
print(find_invalid_positions("(()))(")) # [4, 5]
Pattern 2: Counter Instead of Stack (Only One Bracket Type)
When the problem only involves ( and ), you can replace the stack with a simple integer counter. This drops space complexity from O(n) to O(1):
python example
def is_valid_simple(s):
"""O(1) space for single bracket type problems."""
balance = 0
for c in s:
if c == '(':
balance += 1
elif c == ')':
balance -= 1
if balance < 0:
return False # More ')' than '(' so far
return balance == 0Pattern 3: Two-Pass for Complex Variants
For minimum removal problems, a forward pass handles unmatched ), and a backward pass handles unmatched (:
Example
def min_remove_to_make_valid(s):
"""Remove minimum chars to make parentheses valid."""
s = list(s)
# Forward pass: mark unmatched ')'
open_count = 0
for i, c in enumerate(s):
if c == '(':
open_count += 1
elif c == ')':
if open_count > 0:
open_count -= 1
else:
s[i] = '' # Mark for removal
# Backward pass: mark excess '('
close_count = 0
for i in range(len(s) - 1, -1, -1):
if s[i] == '(':
if close_count > 0:
close_count -= 1
else:
s[i] = '' # Mark for removal
elif s[i] == ')':
close_count += 1
return ''.join(s)
print(min_remove_to_make_valid("lee(t(c)o)de)")) # "lee(t(c)o)de"
print(min_remove_to_make_valid("a)b(c)d")) # "ab(c)d"
Common Edge Cases to Test
Always test your bracket validator against these inputs:
| Input | Expected | Why |
|---|---|---|
"" | Valid | Empty string is valid |
"(" | Invalid | Unclosed bracket |
")" | Invalid | Unmatched close |
"()" | Valid | Single pair |
"([{}])" | Valid | Nested, different types |
"([)]" | Invalid | Wrong nesting order |
"(((" | Invalid | Three unclosed |
"))((" | Invalid | Wrong order overall |
"a(b)c" | Valid | Letters mixed in |
Summary
Balanced parentheses problems are solved with a stack because:
- Opening brackets are pushed (remembered for later matching)
- Closing brackets trigger a pop and comparison with the top
- Stack empty at end = all brackets matched
Variants to know:
- Basic validation: O(n) time, O(n) space
- Minimum additions: O(n) time, O(1) space (count unmatched)
- Longest valid substring: O(n) time, O(n) space (track indices)
- Score: O(n) time, O(n) space (track running sums per level)
- Generate all: O(4^n/√n) time, backtracking
- Remove invalid: O(2^n) time, BFS level by level
The counter-based O(1) space solution works when only one bracket type is involved. For multiple bracket types, a stack is essential.
*Next Lesson: Infix Prefix Postfix*
Continue the lesson path
Study next in this module
01StacksInfix, Prefix, and Postfix Expressions — Stack-Based Conversion and EvaluationMaster expression notation conversion using stacks — infix to postfix (Shunting Yard algorithm), infix to prefix, postfix evaluation, prefix evaluation, why postfix and prefix are easier to evaluate than infix, operator precedence handling, and complete traced examples.02StacksStack Implementation — Three Complete Implementations with Edge CasesComplete implementations of stacks in three ways — dynamic array (Python list), fixed-size array, and singly linked list. Each implementation handles all edge cases with error checking, includes performance analysis, and discusses when to choose each approach.03StacksStack Interview Problems — 10 Classic Problems with Complete SolutionsSolve 10 classic stack interview problems with full solutions — valid parentheses, evaluate reverse polish notation, daily temperatures, next greater element, asteroid collision, largest rectangle in histogram, remove k digits, decode string, and more.04StacksStack — Complete Introduction to LIFO Data StructureA thorough introduction to stacks — the LIFO (Last In, First Out) data structure. Learn the stack model, all operations with complexity, real-world applications, stack implementations using arrays and linked lists, and the fundamental programming constructs that depend on stacks.